fOptions (version 251.70)

BarrierOptions: Valuation of Barrier Options

Description

A collection and description of functions to valuate barrier options. Barrier options are path-dependent options, with payoffs that depend on the price of the underlying asset at expiration and whether or not the asset price crosses a barrier during the life of the option. There are two categories or types of Barrier options: "knock-in" and "knock-out". "Knock-in" or "in" options are paid for up front, but you do not receive the option until the asset price crosses the barrier. "Knock-out" or "out" options come into existence on the issue date but becomes worthless if the asset price hits the barrier before the expiration date. If the option is a knock-in (knock-out), a predetermined cash rebate may be paid at expiration if the option has not been knocked in (knocked-out) during its lifetime. The barrier monitoring frequency specifies how often the price is checked for a breach of the barrier. All of the analytical models have a flag to change the monitoring frequency where the default frequency is continuous. The functions are:

ll{ StandardBarrierOption Standard Barrier Option, DoubleBarrierOption Double Barrier Option, PTSingleAssetBarrierOption Partial Time Single Asset Barrier Option, TwoAssetBarrierOption Two Asset Barrier Option, PTTwoAssetBarrierOption Partial Time Two Asset Barrier Option, LookBarrierOption Lookback Barrier Option, DiscreteBarrierOption Discrete Barrier Option SoftBarrierOption Soft Barrier Option. }

Usage

StandardBarrierOption(TypeFlag, S, X, H, K, Time, r, b, sigma,
    title = NULL, description = NULL)
DoubleBarrierOption(TypeFlag, S, X, L, U, Time, r, b, sigma, 
    delta1, delta2, title = NULL, description = NULL)
PTSingleAssetBarrierOption(TypeFlag, S, X, H, time1, Time2, r, 
    b, sigma, title = NULL, description = NULL)
TwoAssetBarrierOption(TypeFlag, S1, S2, X, H, Time, r, b1, b2,
    sigma1, sigma2, rho, title = NULL, description = NULL)
PTTwoAssetBarrierOption(TypeFlag, S1, S2, X, H, time1, Time2, r, 
    b1, b2, sigma1, sigma2, rho, title = NULL, description = NULL)
LookBarrierOption(TypeFlag, S, X, H, time1, Time2, r, b, sigma,
    title = NULL, description = NULL)
DiscreteBarrierOption(S, H, sigma, dt, title = NULL, 
    description = NULL)
SoftBarrierOption(TypeFlag, S, X, L, U, Time, r, b, sigma, 
    title = NULL, description = NULL)

Arguments

b

the annualized cost-of-carry rate, a numeric value; e.g. 0.1 means 10% pa.

b1, b2

[PTTwoAssetBarrier*] - the annualized cost-of-carry rate for the first and second asset, a numeric value.

delta1, delta2

[DoubleBarrier*] - numeric values which determine the curvature of the lower L and upper U bounds. The case of delta1=delta2=0 corresponds to two flat boundaries, delta1<0 correponds t

description

a character string which allows for a brief description.

dt

[DiscreteBarrier*] - time between monitoring instants, a numeric value.

H

[StandardBarrier*] - the barrier value, a numeric value.

K

[StandardBarrier*] - for an "In"-Barrier a prespecified cash rebate which is paid out at option expiration if the option has not been knocked in during its lifetime, for an "Out"-Barrier a prespecified cash rebate which is paid

L, U

[DoubleBarrier*] - the lower and upper boundary to be touched, numerical values.

r

the annualized rate of interest, a numeric value; e.g. 0.25 means 25% pa.

rho

[TwoAssetBarrier*] - the correlation of the volatility between the first and second asset, a numeric value.

S

the asset price, a numeric value.

S1, S2

[PTTwoAssetBarrier*] - the price of the first and second asset, a numeric value.

sigma

the annualized volatility of the underlying security, a numeric value; e.g. 0.3 means 30% volatility pa.

sigma1, sigma2

[PTTwoAssetBarrier*] - the annualized volatility of the first and second underlying security, numeric values.

Time

the time to maturity measured in years, a numeric value; e.g. 0.5 means 6 months.

time1, Time2

[PTSingleAssetBarrier*][PTTwoAssetBarrier*] - so called type "A" options (see the TypeFlag argument) will have the location of the monitoring period starting at the options starting date and ending at an arbitrary time

title

a character string which allows for a project title.

TypeFlag

usually a character string either "c" for a call option or a "p" for a put option. [StandardBarrier*] - here "cdi" denotes a down-and-in call, "cui" denotes an up-and-in

X

the exercise price, a numeric value.

Value

• The option price, a numeric value.

Details

Single [Standard] Barrier Options: There are four types of single barrier options. The type flag "cdi" denotes a down-and-in call, "cui" denotes an up-and-in call, "cdo" denotes a down-and-out call, and "cuo" denotes an up-and-out call. Similarily, the type flags for the corresponding puts are cdi, cui, cdo, and cuo. A down-and-in option comes into existence and knocked-in only if the asset price falls to the barrier level. An up-and- in option comes into existence and knocked-in only if the asset price rises to the barrier level. A down-and-out option comes into existence and knocked-out only if the asset price falls to the barrier level. An up-and-in option comes into existence and knocked-out only if the asset price rises to the barrier level. European single barrier options can be priced analytically using a model introduced by Reiner and Rubinstein (1991). A trinomial lattice is used for the numerical calculation of an American or European style single barrier options. [Haug's Book, Chapter 2.10.1] Double Barrier Options: A double barrier option is either knocked in or knocked out if the asset price touches the lower or upper barrier during its lifetime. The type flag "co" denotes an up-and-out-down-and-out call, "ci" denotes an up-and-in-down-and-in call, "po" denotes an up-and-out-down-and-out put, and "pi" denotes an up-and-in-down-and-in call. Once a barrier is crossed, the option comes into existence if it is a knock-in barrier or becomes worthless if it is a knocked out barrier. Double barrier options can be priced analytically using a model introduced by Ikeda and Kunitomo (1992). [Haug's Book, Chapter 2.10.2] Partial-Time Barrier Options: For single asset partial-time barrier options, the monitoring period for a barrier crossing is confined to only a fraction of the option's lifetime. There are two types of partial-time barrier options: partial-time-start and partial-time-end. Partial-time-start barrier options have the monitoring period start at time zero and end at an arbitrary date before expiration. Partial-time-end barrier options have the monitoring period start at an arbitrary date before expiration and end at expiration. Partial-time-end barrier options are then broken down again into two categories: B1 and B2. Type B1 is defined such that only a barrier hit or crossed causes the option to be knocked out. There is no difference between up and down options. Type B2 options are defined such that a down-and-out call is knocked out as soon as the underlying price is below the barrier. Similarly, an up-and-out call is knocked out as soon as the underlying price is above the barrier. Partial-time barrier options can be priced analytically using a model introduced by Heynen and Kat (1994). [Haug's Book, Chapter 2.10.3] Two-Asset Barrier Options: The underlying asset, Asset 1, determines how much the option is in or out-of-the-money. The other asset, Asset 2, is the trigger asset that is linked to barrier hits. Two-asset barrier options can be priced analytically using a model introduced by Heynen and Kat (1994). [Haug's Book, Chapter 2.10.4] Lookback Barrier Options: A look-barrier option is the combination of a forward starting fixed strike Lookback option and a partial time barrier option. The option's barrier monitoring period starts at time zero and ends at an arbitrary date before expiration. If the barrier is not triggered during this period, the fixed strike Lookback option will be kick off at the end of the barrier tenor. Lookback barrier options can be priced analytically using a model introduced by Bermin (1996). [Haug's Book, Chapter 2.10.6] Partial-Time-Two-Asset Options: Partial-time two-asset barrier options are similar to standard two-asset barrier options, except that the barrier hits are monitored only for a fraction of the option's lifetime. The option is knocked in or knocked out is Asset 2 hits the barrier during the monitoring period. The payoff depends on Asset 1 and the strike price. Partial-time two-asset barrier options can be priced analytically using a model introduced by Bermin (1996). [Haug's Book, Chapter 2.10.5] Soft Barrier Options: A soft-barrier option is similar to a standard barrier option, except that the barrier is no longer a single level. Rather, it is a soft range between a lower level and an upper level. Soft-barrier options are knocked in or knocked out proportionally. Introduced by Hart and Ross (1994), the valuation formula can be used to price soft-down-and-in call and soft-up-and-in put options. The value of the related "out" option can be determined by subtracting the "in" option value from the value of a standard plain option. Soft-barrier options can be priced analytically using a model introduced by Hart and Ross (1994). [Haug's Book, Chapter 2.10.8]

References

Haug E.G. (1997); The complete Guide to Option Pricing Formulas, Chapter 2.10, McGraw-Hill, New York.

Examples

## Examples from Chapter 2.10 in E.G. Haug's Option Guide (1997)

## Standard Barrier Option [2.10.1]:
   # down-and-out Barrier Call
   StandardBarrierOption(TypeFlag = "cdo", S = 100, X = 90, 
     H = 95, K = 3, Time = 0.5, r = 0.08, b = 0.04, sigma = 0.25)
   
## Double Barrier Option [2.10.2]:
   DoubleBarrierOption(TypeFlag = "co", S = 100, X = 100, L = 50, 
     U = 150, Time = 0.25, r = 0.10, b = 0.10, sigma = 0.15, 
     delta1 = -0.1, delta2 = 0.1)
   
## Partial Time Single-Asset Barrier Option [2.10.3]:
   PTSingleAssetBarrierOption(TypeFlag = "coB1", S = 95, X = 110, 
     H = 100, time1 = 0.5, Time2 = 1, r = 0.20, b = 0.20, 
     sigma = 0.25)
   
## Two Asset Barrier Option [2.10.4]:
   TwoAssetBarrierOption(TypeFlag = "puo", S1 = 100, S2 = 100, 
     X = 110, H = 105, Time = 0.5, r = 0.08, b1 = 0.08, b2 = 0.08, 
     sigma1 = 0.2, sigma2 = 0.2, rho = -0.5)
   
## PT Two Asset Barrier Option [2.10.5]:
   PTTwoAssetBarrierOption(TypeFlag = "pdo", S1 = 100, S2 = 100, 
     X = 100, H = 85, time1 = 0.5, Time2 = 1, r = 0.1, b1 = 0.1, 
     b2 = 0.1, sigma1 = 0.25, sigma2 = 0.30, rho = -0.5)
   
## Look Barrier Option [2.10.6]:
   LookBarrierOption(TypeFlag = "cuo", S = 100, X = 100, H = 130, 
     time1 = 0.25, Time2 = 1, r = 0.1, b = 0.1, sigma = 0.15)
   LookBarrierOption(TypeFlag = "cuo", S = 100, X = 100, H = 110, 
     time1 = 1, Time2 = 1, r = 0.1, b = 0.1, sigma = 0.30)
   
## Discrete Barrier Option [2.10.7]:  
   DiscreteBarrierOption(S = 100, H = 105, sigma = 0.25, dt = 0.1)
   
## Soft Barrier Option [2.10.8]:
   SoftBarrierOption(TypeFlag = "cdo", S = 100, X = 100, L = 70, 
     U = 95, Time = 0.5, r = 0.1, b = 0.05, sigma = 0.20)